WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = [1] x1 + [5]
p(0) = [8]
p(div) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(if) = [1] x1 + [2] x2 + [1] x3 + [4]
p(lt) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
-(x,0()) = [1] x + [5]
> [1] x + [0]
= x
-(0(),s(y)) = [13]
> [8]
= 0()
if(false(),x,y) = [2] x + [1] y + [4]
> [1] y + [0]
= y
if(true(),x,y) = [2] x + [1] y + [4]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
-(s(x),s(y)) = [1] x + [5]
>= [1] x + [5]
= -(x,y)
div(x,0()) = [1] x + [8]
>= [8]
= 0()
div(0(),y) = [1] y + [8]
>= [8]
= 0()
div(s(x),s(y)) = [1] x + [1] y + [0]
>= [1] x + [1] y + [25]
= if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(x,0()) = [0]
>= [0]
= false()
lt(0(),s(y)) = [0]
>= [0]
= true()
lt(s(x),s(y)) = [0]
>= [0]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = [1] x1 + [12]
p(0) = [1]
p(div) = [1] x1 + [2] x2 + [1]
p(false) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(lt) = [0]
p(s) = [1] x1 + [1]
p(true) = [13]
Following rules are strictly oriented:
-(s(x),s(y)) = [1] x + [13]
> [1] x + [12]
= -(x,y)
div(x,0()) = [1] x + [3]
> [1]
= 0()
div(0(),y) = [2] y + [2]
> [1]
= 0()
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [12]
>= [1] x + [0]
= x
-(0(),s(y)) = [13]
>= [1]
= 0()
div(s(x),s(y)) = [1] x + [2] y + [4]
>= [1] x + [2] y + [18]
= if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) = [1] x + [1] y + [2]
>= [1] y + [0]
= y
if(true(),x,y) = [1] x + [1] y + [14]
>= [1] x + [0]
= x
lt(x,0()) = [0]
>= [1]
= false()
lt(0(),s(y)) = [0]
>= [13]
= true()
lt(s(x),s(y)) = [0]
>= [0]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = [1] x1 + [11]
p(0) = [6]
p(div) = [1] x1 + [4] x2 + [0]
p(false) = [0]
p(if) = [1] x1 + [3] x2 + [1] x3 + [0]
p(lt) = [2]
p(s) = [1] x1 + [0]
p(true) = [1]
Following rules are strictly oriented:
lt(x,0()) = [2]
> [0]
= false()
lt(0(),s(y)) = [2]
> [1]
= true()
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [11]
>= [1] x + [0]
= x
-(0(),s(y)) = [17]
>= [6]
= 0()
-(s(x),s(y)) = [1] x + [11]
>= [1] x + [11]
= -(x,y)
div(x,0()) = [1] x + [24]
>= [6]
= 0()
div(0(),y) = [4] y + [6]
>= [6]
= 0()
div(s(x),s(y)) = [1] x + [4] y + [0]
>= [1] x + [4] y + [31]
= if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) = [3] x + [1] y + [0]
>= [1] y + [0]
= y
if(true(),x,y) = [3] x + [1] y + [1]
>= [1] x + [0]
= x
lt(s(x),s(y)) = [2]
>= [2]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) = [1] x1 + [0]
p(0) = [0]
p(div) = [2] x1 + [0]
p(false) = [1]
p(if) = [2] x1 + [2] x2 + [1] x3 + [2]
p(lt) = [1]
p(s) = [1] x1 + [8]
p(true) = [1]
Following rules are strictly oriented:
div(s(x),s(y)) = [2] x + [16]
> [2] x + [12]
= if(lt(x,y),0(),s(div(-(x,y),s(y))))
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
-(0(),s(y)) = [0]
>= [0]
= 0()
-(s(x),s(y)) = [1] x + [8]
>= [1] x + [0]
= -(x,y)
div(x,0()) = [2] x + [0]
>= [0]
= 0()
div(0(),y) = [0]
>= [0]
= 0()
if(false(),x,y) = [2] x + [1] y + [4]
>= [1] y + [0]
= y
if(true(),x,y) = [2] x + [1] y + [4]
>= [1] x + [0]
= x
lt(x,0()) = [1]
>= [1]
= false()
lt(0(),s(y)) = [1]
>= [1]
= true()
lt(s(x),s(y)) = [1]
>= [1]
= lt(x,y)
* Step 5: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) = [1 0] x1 + [0]
[0 1] [0]
p(0) = [0]
[0]
p(div) = [2 1] x1 + [2 0] x2 + [7]
[0 1] [0 0] [0]
p(false) = [0]
[2]
p(if) = [4 0] x1 + [2 4] x2 + [1 0] x3 + [2]
[0 0] [2 2] [0 1] [0]
p(lt) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 2] [2]
p(s) = [1 4] x1 + [1]
[0 1] [1]
p(true) = [0]
[4]
Following rules are strictly oriented:
lt(s(x),s(y)) = [0 1] x + [0 0] y + [1]
[0 0] [0 2] [4]
> [0 1] x + [0 0] y + [0]
[0 0] [0 2] [2]
= lt(x,y)
Following rules are (at-least) weakly oriented:
-(x,0()) = [1 0] x + [0]
[0 1] [0]
>= [1 0] x + [0]
[0 1] [0]
= x
-(0(),s(y)) = [0]
[0]
>= [0]
[0]
= 0()
-(s(x),s(y)) = [1 4] x + [1]
[0 1] [1]
>= [1 0] x + [0]
[0 1] [0]
= -(x,y)
div(x,0()) = [2 1] x + [7]
[0 1] [0]
>= [0]
[0]
= 0()
div(0(),y) = [2 0] y + [7]
[0 0] [0]
>= [0]
[0]
= 0()
div(s(x),s(y)) = [2 9] x + [2 8] y + [12]
[0 1] [0 0] [1]
>= [2 9] x + [2 8] y + [12]
[0 1] [0 0] [1]
= if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) = [2 4] x + [1 0] y + [2]
[2 2] [0 1] [0]
>= [1 0] y + [0]
[0 1] [0]
= y
if(true(),x,y) = [2 4] x + [1 0] y + [2]
[2 2] [0 1] [0]
>= [1 0] x + [0]
[0 1] [0]
= x
lt(x,0()) = [0 1] x + [0]
[0 0] [2]
>= [0]
[2]
= false()
lt(0(),s(y)) = [0 0] y + [0]
[0 2] [4]
>= [0]
[4]
= true()
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))